In an attempt to prove Bondy and Erdös conjecture \(r(C_n,K_m) = (n-1)(m-1)+1\) , for all \((n,m)\ne (3,3)\)  satisfying \(n \ge m \ge 3\)  under certain restrictions, Schiermeyer has proved that \(r(C_n,K_6) = 5(n - 1)+1\) , for \(n \ge 6\)  (see [1]}, [2]}). Characterizing all Ramsey critical \((C_n,K_6)\)  graphs boils down to finding all (red/blue) colorings of \(K_{r(C_n,K_6)-1}\)  such that there is no red \(C_n\)  or a blue \(K_6\) . This is achieved by finding all \(C_n\) -free graphs on \(K_{r(C_n,K_6)-1}\)  vertices such that \(\alpha (G)<6\) . We first prove that any \(C_n\)  -free graph (where \(n \ge 15\) ) of order \(5(n-1)\)  with \(\alpha (G) \le 5\)  contains a \(5K_{n-1}\) . To prove this, we use seven lemmas of which the first three are already proven results. For ease of reference, we reiterate Lemma REF  from [3]}, Lemma REF  from [4]} and Lemma REF  by Bollobás et al from [5]}.
